OCR C4 (Core Mathematics 4) Specimen

1 Find the quotient and remainder when \(x ^ { 4 } + 1\) is divided by \(x ^ { 2 } + 1\).

2

1. Expand \(( 1 - 2 x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
2. State the set of values for which the expansion in part (i) is valid.

3 Find \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), giving your answer in terms of e.

4
\includegraphics[max width=\textwidth, alt={}, center]{798da17d-0af5-4aa6-b731-564642dc28d5-2_428_572_861_760} As shown in the diagram the points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) with respect to the origin \(O\).

1. Make a sketch of the diagram, and mark the points \(C , D\) and \(E\) such that \(\overrightarrow { O C } = 2 \mathbf { a } , \overrightarrow { O D } = 2 \mathbf { a } + \mathbf { b }\) and \(\overrightarrow { O E } = \frac { 1 } { 3 } \overrightarrow { O D }\).
2. By expressing suitable vectors in terms of \(\mathbf { a }\) and \(\mathbf { b }\), prove that \(E\) lies on the line joining \(A\) and \(B\).

5

1. For the curve \(2 x ^ { 2 } + x y + y ^ { 2 } = 14\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Deduce that there are two points on the curve \(2 x ^ { 2 } + x y + y ^ { 2 } = 14\) at which the tangents are parallel to the \(x\)-axis, and find their coordinates.

6
\includegraphics[max width=\textwidth, alt={}, center]{798da17d-0af5-4aa6-b731-564642dc28d5-3_766_611_251_703} The diagram shows the curve with parametric equations $$x = a \sin \theta , \quad y = a \theta \cos \theta$$ where \(a\) is a positive constant and \(- \pi \leqslant \theta \leqslant \pi\). The curve meets the positive \(y\)-axis at \(A\) and the positive \(x\)-axis at \(B\).

1. Write down the value of \(\theta\) corresponding to the origin, and state the coordinates of \(A\) and \(B\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \theta \tan \theta\), and hence find the equation of the tangent to the curve at the origin.

7 The line \(L _ { 1 }\) passes through the point \(( 3,6,1 )\) and is parallel to the vector \(2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k }\). The line \(L _ { 2 }\) passes through the point ( \(3 , - 1,4\) ) and is parallel to the vector \(\mathbf { i } - 2 \mathbf { j } + \mathbf { k }\).

1. Write down vector equations for the lines \(L _ { 1 }\) and \(L _ { 2 }\).
2. Prove that \(L _ { 1 }\) and \(L _ { 2 }\) intersect, and find the coordinates of their point of intersection.
3. Calculate the acute angle between the lines.

8 Let \(I = \int \frac { 1 } { x ( 1 + \sqrt { } x ) ^ { 2 } } \mathrm {~d} x\).

1. Show that the substitution \(u = \sqrt { } x\) transforms \(I\) to \(\int \frac { 2 } { u ( 1 + u ) ^ { 2 } } \mathrm {~d} u\).
2. Express \(\frac { 2 } { u ( 1 + u ) ^ { 2 } }\) in the form \(\frac { A } { u } + \frac { B } { 1 + u } + \frac { C } { ( 1 + u ) ^ { 2 } }\).
3. Hence find \(I\).

9
\includegraphics[max width=\textwidth, alt={}, center]{798da17d-0af5-4aa6-b731-564642dc28d5-4_572_917_294_607} A cylindrical container has a height of 200 cm . The container was initially full of a chemical but there is a leak from a hole in the base. When the leak is noticed, the container is half-full and the level of the chemical is dropping at a rate of 1 cm per minute. It is required to find for how many minutes the container has been leaking. To model the situation it is assumed that, when the depth of the chemical remaining is \(x \mathrm {~cm}\), the rate at which the level is dropping is proportional to \(\sqrt { } x\). Set up and solve an appropriate differential equation, and hence show that the container has been leaking for about 80 minutes.